In this and the previous case the likelihood function takes the form of a prod­uct. However, in the dependent case we can also write the likelihood function as a product. For example, let Z = (ZT, Zj)T be absolutely continuously distributed with joint density fn (zn,…, zi в0), where the Zj’s are no longer independent. It is always possible to decompose a joint density as a product of conditional densities and an initial marginal density. In particular, letting, for t > 2,

ft (ztzt-1, …, Zl, в) = ft (zt, …, z1 в )/ft-1(zt-1, …, z1 в),

we can write

n

fn (zn, Zlв) = fi(zie )f[ ft (zt zt-1, …,Z1,0).

t =2

Therefore, the likelihood function in this case can be written as

n

L n (в) = fn (Zn, Zi ) = fi( Z ів Щ f (Zt Zt-i, Z і, в).

t =2

(8.6)

It is easy to verify that in this case (8.5) also holds, and therefore so does (8.3). Moreover, it follows straightforwardly from (8.6) and the preceding argument that